- Split input into 2 regimes
if x < -1.86902872977817849e-4
Initial program 0.0
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
- Using strategy
rm Applied flip--0.0
\[\leadsto \sqrt{\frac{\color{blue}{\frac{e^{2 \cdot x} \cdot e^{2 \cdot x} - 1 \cdot 1}{e^{2 \cdot x} + 1}}}{e^{x} - 1}}\]
Simplified0.0
\[\leadsto \sqrt{\frac{\frac{\color{blue}{\left(-1 \cdot 1\right) + {\left(e^{2}\right)}^{\left(2 \cdot x\right)}}}{e^{2 \cdot x} + 1}}{e^{x} - 1}}\]
if -1.86902872977817849e-4 < x
Initial program 33.9
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Taylor expanded around 0 6.2
\[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
Simplified6.2
\[\leadsto \color{blue}{0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
- Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l}
\mathbf{if}\;x \le -1.86902872977817849 \cdot 10^{-4}:\\
\;\;\;\;\sqrt{\frac{\frac{\left(-1 \cdot 1\right) + {\left(e^{2}\right)}^{\left(2 \cdot x\right)}}{e^{2 \cdot x} + 1}}{e^{x} - 1}}\\
\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\
\end{array}\]
